Persistent homology is the mathematical language to describe components and
holes of different dimensions and scales in a manifold (or its generalization). Homology, a concept in algebraic topology, captures the number of components and holes in a manifold
with linear algebra and group theory.
Persistent homology generalizes homology and keeps track of
the evolution of these components and holes as the resolution changes.
Such information may be succinctly summarized in persistence diagrams. Since all
these concepts are based on algebra, they are amenable to algorithmic computation.
Below we explain these concepts with examples. We refer the reader to
[1]} for an exposition on algebraic
topology, [2]} for a gentle introduction to homology theory, [3]}, for an exposition on persistent homology, and
[4]}, [5]} for short surveys on
topological data analysis.
